You don't have to raise the limper with QJs out of position if you don't want too, just checking and playing postflop is fine as well. If you do raise, which means you think you can outmaneuver that fishy player postflop most of the time, you want more money into the pot so you can win more money postflop. Make it 6bb (4bb + 1bb for the limper + 1bb for being out of position ).

As played the continuation bet is fine on that flop, it's not that drawy, and it will miss your opponent often if he plays mostly high cards, which bad players like to do. Plus even if you get called you have 2 overcards and some backdoor draws.

On the turn you pick up 8 more outs, and you opponent didn't really show strength by just calling the flop. He could have Tx, 7x, any other pocket pair, broadway cards that he didn't want to fold, possibly 98 for a straight draw. I doubt you will be able to get him off 98, 88 and Tx, but you can probably fold out all the rest by betting again, and you still likely have 14 outs to improve to the best hand, tho you need to be careful on some rivers, especially a J. Betting again here is fine, i'd probably make it a bit bigger, so that MP2 has less than a potsize bet left on the river and is thus more inclined to call a shove from us on the river with whatever hand he has when we do improve.